Short Answer 
The problem starts by defining variables for pancakes (x) and waffles (y), with y initially at 80. It sets up an equation considering changes due to baking and eating, ultimately confirming that there are still 80 waffles after the adjustments.
Step 1: Define Variables
To solve the problem, we start by defining our variables. Let x represent the number of pancakes and y the number of waffles. Initially, we have:
- Let x = number of pancakes
- Let y = number of waffles
We know that initially, there are 80 waffles, so y = 80.
Step 2: Set Up Equations
Next, we need to set up our equations based on the problem’s conditions. After baking more waffles and the grandchildren eating pancakes, we can express the new quantities:
- Waffles: y + 45
- Pancakes: x – 12
From the problem, we also know that there are 17 more waffles than pancakes after these changes, leading us to the equation:
- y + 45 = x – 12 + 17 (Equation 1)
Step 3: Solve for the Number of Waffles
To find the number of waffles, we substitute x in Equation 1 using the relationship x = 1.5y. This gives us:
- y + 45 = 1.5y – 12 + 17
- Rearranging: 0.5y = 40
- Now, solving for y: y = 40 / 0.5 = 80 waffles
Thus, we confirm that the initial number of waffles is indeed 80.